
theorem Th18: ::p.122 lemma 3.4.(ii)
  for T being T_0-TopSpace ex M being non empty set, f being
  Function of T, product (M --> Sierpinski_Space) st corestr(f) is
  being_homeomorphism
proof
  let T be T_0-TopSpace;
  take M = the carrier of (InclPoset the topology of T);
  set J = M --> Sierpinski_Space;
  reconsider PP = the set of all
product((Carrier J)+*(m,{1})) where m is Element of M
 as prebasis of product J by Th13;
  deffunc F(object) =
  chi({u where u is Subset of T: $1 in u & u is open}, the topology of T);
  consider f being Function such that
A1: dom f = the carrier of T and
A2: for x being object st x in the carrier of T holds f.x = F(x) from
  FUNCT_1:sch 3;
  rng f c= the carrier of product J
  proof
    let y be object;
    assume y in rng f;
    then consider x being object such that
A3: x in dom f & y = f.x by FUNCT_1:def 3;
    set ch = chi({u where u is Subset of T: x in u & u is open}, the topology
    of T);
A4: dom ch = the topology of T by FUNCT_3:def 3
      .= M by YELLOW_1:1
      .= dom (Carrier J) by PARTFUN1:def 2;
A5: for z being object st z in dom (Carrier J) holds ch.z in (Carrier J).z
    proof
      let z be object;
      assume z in dom (Carrier J);
      then
A6:   z in M;
      then z in the topology of T by YELLOW_1:1;
      then z in dom ch by FUNCT_3:def 3;
      then
A7:   ch.z in rng ch by FUNCT_1:def 3;
      ex R being 1-sorted st R = J.z & (Carrier J).z = the carrier of R by A6,
PRALG_1:def 15;
      then (Carrier J).z = the carrier of Sierpinski_Space by A6,FUNCOP_1:7
        .= {0,1} by Def9;
      hence thesis by A7;
    end;
    y = ch by A1,A2,A3;
    then y in product Carrier J by A4,A5,CARD_3:def 5;
    hence thesis by Def3;
  end;
  then reconsider f as Function of T, product J by A1,FUNCT_2:def 1,RELSET_1:4;
  take f;
A8: rng (corestr f) = [#](Image f) by FUNCT_2:def 3;
  for A being Subset of product J st A in PP holds f"A is open
  proof
    {1} c= {0,1} by ZFMISC_1:7;
    then reconsider V = {1} as Subset of Sierpinski_Space by Def9;
    let A be Subset of product J;
    reconsider a = A as Subset of product Carrier J by Def3;
    assume A in PP;
    then consider i being Element of M such that
A9: a = product ((Carrier J) +* (i,{1}));
A10: i in M;
    then
A11: i in the topology of T by YELLOW_1:1;
    then reconsider i9 = i as Subset of T;
A12: i in dom (Carrier J) by A10,PARTFUN1:def 2;
A13: i c= f"A
    proof
      let z be object;
      assume
A14:  z in i;
      set W = {u where u is Subset of T: z in u & u is open};
      i9 is open by A11;
      then
A15:  i in W by A14;
A16:  for w being object st w in dom ((Carrier J) +* (i,V)) holds chi(W,the
      topology of T).w in ((Carrier J) +* (i,V)).w
      proof
        let w be object;
        assume w in dom ((Carrier J) +* (i,V));
        then w in dom (Carrier J) by FUNCT_7:30;
        then
A17:    w in M;
        then w in the topology of T by YELLOW_1:1;
        then
A18:    w in dom chi(W,the topology of T) by FUNCT_3:def 3;
        per cases;
        suppose
          w = i;
          then
          ((Carrier J) +* (i,V)).w = {1} & chi(W,the topology of T).w = 1
          by A11,A12,A15,FUNCT_3:def 3,FUNCT_7:31;
          hence thesis by TARSKI:def 1;
        end;
        suppose
A19:      w <> i;
A20:      chi(W,the topology of T).w in rng chi(W,the topology of T ) by A18,
FUNCT_1:def 3;
          ex r being 1-sorted st r = J.w & (Carrier J).w = the carrier of
          r by A17,PRALG_1:def 15;
          then
A21:      (Carrier J).w = the carrier of Sierpinski_Space by A17,FUNCOP_1:7
            .= {0,1} by Def9;
          ((Carrier J) +* (i,V)).w = (Carrier J).w by A19,FUNCT_7:32;
          hence thesis by A21,A20;
        end;
      end;
A22:  dom chi(W,the topology of T) = the topology of T by FUNCT_3:def 3
        .= M by YELLOW_1:1
        .= dom (Carrier J) by PARTFUN1:def 2
        .= dom ((Carrier J) +* (i,V)) by FUNCT_7:30;
A23:  z in i9 by A14;
      then f.z = chi(W,the topology of T) by A2;
      then f.z in A by A9,A22,A16,CARD_3:def 5;
      hence thesis by A1,A23,FUNCT_1:def 7;
    end;
A24: ((Carrier J)+*(i,V)).i = {1} by A12,FUNCT_7:31;
    f"A c= i
    proof
      let z be object;
      set W = {u where u is Subset of T: z in u & u is open};
      assume z in f"A;
      then f.z in a & f.z = chi(W,the topology of T) by A2,FUNCT_1:def 7;
      then consider g being Function such that
A25:  chi(W,the topology of T) = g and
      dom g = dom ((Carrier J) +* (i,V)) and
A26:  for w being object st w in dom ((Carrier J) +* (i,V)) holds g.w in
      ((Carrier J) +* (i,V)).w by A9,CARD_3:def 5;
      i in dom ((Carrier J) +* (i,V)) by A12,FUNCT_7:30;
      then g.i in ((Carrier J) +* (i,V)).i by A26;
      then chi(W,the topology of T).i = 1 by A24,A25,TARSKI:def 1;
      then i in W by FUNCT_3:36;
      then ex uu being Subset of T st i = uu & z in uu & uu is open;
      hence thesis;
    end;
    then f"A = i by A13;
    hence thesis by A11;
  end;
  then f is continuous by YELLOW_9:36;
  then
A27: dom (corestr f) = [#]T & corestr f is continuous by Th10,FUNCT_2:def 1;
A28: corestr f is one-to-one
  proof
    let x,y be Element of T;
    set U1 = {u where u is Subset of T: x in u & u is open};
    set U2 = {u where u is Subset of T: y in u & u is open};
    assume
A29: (corestr f).x = (corestr f).y;
    thus x = y
    proof
A30:  f.x = chi(U1,the topology of T) & f.y = chi(U2,the topology of T) by A2;
      assume not thesis;
      then consider V being Subset of T such that
A31:  V is open and
A32:  x in V & not y in V or y in V & not x in V by T_0TOPSP:def 7;
A33:  V in the topology of T by A31;
      per cases by A32;
      suppose
A34:    x in V & not y in V;
        reconsider v = V as Subset of T;
A35:    not v in U2
        proof
          assume not thesis;
          then ex u being Subset of T st u = v & y in u & u is open;
          hence thesis by A34;
        end;
        v in U1 by A31,A34;
        then chi(U1,the topology of T).v = 1 by A33,FUNCT_3:def 3;
        hence thesis by A29,A30,A33,A35,FUNCT_3:def 3;
      end;
      suppose
A36:    y in V & not x in V;
        reconsider v = V as Subset of T;
A37:    not v in U1
        proof
          assume not thesis;
          then ex u being Subset of T st u = v & x in u & u is open;
          hence thesis by A36;
        end;
        v in U2 by A31,A36;
        then chi(U2,the topology of T).v = 1 by A33,FUNCT_3:def 3;
        hence thesis by A29,A30,A33,A37,FUNCT_3:def 3;
      end;
    end;
  end;
A38: for V being Subset of T st V is open holds f.:V = product ((Carrier J)
  +* (V,{1})) /\ the carrier of Image f
  proof
    let V be Subset of T;
    assume
A39: V is open;
    hereby
      let y be object;
      assume y in f.:V;
      then consider x being object such that
A40:  x in dom f and
A41:  x in V and
A42:  y = f.x by FUNCT_1:def 6;
      y in rng f by A40,A42,FUNCT_1:def 3;
      then
A43:  y in the carrier of Image f by Th9;
      set Q = {u where u is Subset of T: x in u & u is open};
A44:  V in Q by A39,A41;
A45:  for W being object st W in dom ((Carrier J) +* (V,{1})) holds chi(Q,
      the topology of T).W in ((Carrier J) +* (V,{1})).W
      proof
        let W be object;
        assume W in dom ((Carrier J) +* (V,{1}));
        then
A46:    W in dom (Carrier J) by FUNCT_7:30;
        then
A47:    W in M;
        then
A48:    W in the topology of T by YELLOW_1:1;
        then
A49:    W in dom chi(Q,the topology of T) by FUNCT_3:def 3;
        per cases;
        suppose
          W = V;
          then
          ((Carrier J) +* (V,{1})).W = {1} & chi(Q,the topology of T).W =
          1 by A44,A46,A48,FUNCT_3:def 3,FUNCT_7:31;
          hence thesis by TARSKI:def 1;
        end;
        suppose
A50:      W <> V;
A51:      chi(Q,the topology of T).W in rng chi(Q,the topology of T) by A49,
FUNCT_1:def 3;
          ex r being 1-sorted st r = J.W & (Carrier J).W = the carrier of
          r by A47,PRALG_1:def 15;
          then
A52:      (Carrier J).W = the carrier of Sierpinski_Space by A47,FUNCOP_1:7
            .= {0,1} by Def9;
          ((Carrier J) +* (V,{1})).W = (Carrier J).W by A50,FUNCT_7:32;
          hence thesis by A52,A51;
        end;
      end;
A53:  dom chi(Q,the topology of T) = the topology of T by FUNCT_3:def 3
        .= M by YELLOW_1:1
        .= dom Carrier J by PARTFUN1:def 2
        .= dom ((Carrier J) +* (V,{1})) by FUNCT_7:30;
      y = chi(Q,the topology of T) by A2,A40,A42;
      then y in product ((Carrier J) +* (V,{1})) by A53,A45,CARD_3:def 5;
      hence y in product ((Carrier J) +* (V,{1})) /\ the carrier of Image f by
A43,XBOOLE_0:def 4;
    end;
    let y be object;
    assume
A54: y in product ((Carrier J) +* (V,{1})) /\ the carrier of Image f;
    then y in product ((Carrier J) +* (V,{1})) by XBOOLE_0:def 4;
    then consider g being Function such that
A55: y = g and
    dom g = dom ((Carrier J) +* (V,{1})) and
A56: for W being object st W in dom ((Carrier J) +* (V,{1})) holds g.W
    in ((Carrier J) +* (V,{1})).W by CARD_3:def 5;
    rng f = the carrier of Image f by Th9;
    then y in rng f by A54,XBOOLE_0:def 4;
    then consider x being object such that
A57: x in dom f & y = f.x by FUNCT_1:def 3;
    V in the topology of T by A39;
    then V in M by YELLOW_1:1;
    then
A58: V in dom (Carrier J) by PARTFUN1:def 2;
    then V in dom ((Carrier J) +* (V,{1})) by FUNCT_7:30;
    then g.V in ((Carrier J) +* (V,{1})).V by A56;
    then
A59: g.V in {1} by A58,FUNCT_7:31;
    set Q = {u where u is Subset of T: x in u & u is open};
    y = chi(Q,the topology of T) by A2,A57;
    then chi(Q,the topology of T).V = 1 by A55,A59,TARSKI:def 1;
    then V in Q by FUNCT_3:36;
    then ex u being Subset of T st u = V & x in u & u is open;
    hence thesis by A57,FUNCT_1:def 6;
  end;
A60: for V being Subset of T st V is open holds ((corestr f)")"V is open
  proof
    let V be Subset of T;
A61: PP c= the topology of product J by TOPS_2:64;
    assume
A62: V is open;
    then V in the topology of T;
    then reconsider W = V as Element of M by YELLOW_1:1;
A63: product((Carrier J)+*(W,{1})) in PP;
    then reconsider Q = product((Carrier J)+*(V,{1})) as Subset of product J;
    (corestr f).:V = Q /\ [#](Image f) by A38,A62;
    then (corestr f).: V in the topology of Image f by A63,A61,PRE_TOPC:def 4;
    then (corestr f).:V is open;
    hence thesis by A8,A28,TOPS_2:54;
  end;
  [#]T <> {};
  then (corestr f)" is continuous by A60,TOPS_2:43;
  hence thesis by A8,A28,A27,TOPS_2:def 5;
end;
